{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:ZLHTANQBMBJYUD2DHGUKF6JAGK","short_pith_number":"pith:ZLHTANQB","canonical_record":{"source":{"id":"1004.2687","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-04-15T18:51:51Z","cross_cats_sorted":["math-ph","math.AT","math.MP"],"title_canon_sha256":"ef374625b534d69b54fdce8bcb83ecb892b455d313ceed8f88f275b810c6ff4e","abstract_canon_sha256":"f4c323d90815eae92fe484b859f1119678a7566c62860af42c9199489f10b7f8"},"schema_version":"1.0"},"canonical_sha256":"cacf30360160538a0f4339a8a2f92032b3267cec8c0dad64a27af60a3cf8b707","source":{"kind":"arxiv","id":"1004.2687","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.2687","created_at":"2026-05-18T04:22:48Z"},{"alias_kind":"arxiv_version","alias_value":"1004.2687v3","created_at":"2026-05-18T04:22:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.2687","created_at":"2026-05-18T04:22:48Z"},{"alias_kind":"pith_short_12","alias_value":"ZLHTANQBMBJY","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"ZLHTANQBMBJYUD2D","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"ZLHTANQB","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:ZLHTANQBMBJYUD2DHGUKF6JAGK","target":"record","payload":{"canonical_record":{"source":{"id":"1004.2687","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-04-15T18:51:51Z","cross_cats_sorted":["math-ph","math.AT","math.MP"],"title_canon_sha256":"ef374625b534d69b54fdce8bcb83ecb892b455d313ceed8f88f275b810c6ff4e","abstract_canon_sha256":"f4c323d90815eae92fe484b859f1119678a7566c62860af42c9199489f10b7f8"},"schema_version":"1.0"},"canonical_sha256":"cacf30360160538a0f4339a8a2f92032b3267cec8c0dad64a27af60a3cf8b707","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:22:48.847374Z","signature_b64":"L6YtO4n9Sf3+xMffrNk4CK4uRn8z6PFbXgvhKOy02ZJzl5T6VxpReI53ZaldpWcBRH4GUGcFPWPGlDv/L2epBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cacf30360160538a0f4339a8a2f92032b3267cec8c0dad64a27af60a3cf8b707","last_reissued_at":"2026-05-18T04:22:48.846845Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:22:48.846845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.2687","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:22:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l6TETFqlzwWJAVskMTz8AABFuevRx6c0BG+im5YJzFNMO/4vUyMKOyGEUaRsgSrKBuXfXu3i0ETnL+H3cL5hAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T11:07:45.545908Z"},"content_sha256":"377566afc7468ae9b3b7edb580c425fd3d7ef145fadda1087c20a7947538e478","schema_version":"1.0","event_id":"sha256:377566afc7468ae9b3b7edb580c425fd3d7ef145fadda1087c20a7947538e478"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:ZLHTANQBMBJYUD2DHGUKF6JAGK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Witten-Hodge theory on manifolds with boundary and equivariant cohomology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AT","math.MP"],"primary_cat":"math.DG","authors_text":"James Montaldi, Qusay S.A. Al-Zamil","submitted_at":"2010-04-15T18:51:51Z","abstract_excerpt":"We consider a compact, oriented, smooth Riemannian manifold $M$ (with or without boundary) and we suppose $G$ is a torus acting by isometries on $M$. Given $X$ in the Lie algebra and corresponding vector field $X_M$ on $M$, one defines Witten's inhomogeneous coboundary operator $d_{X_M} = d+\\iota_{X_M}: \\Omega_G^\\pm \\to\\Omega_G^\\mp$ (even/odd invariant forms on $M$) and its adjoint $\\delta_{X_M}$. In the 1980s Witten showed that the resulting cohomology classes have $X_M$-harmonic representatives (forms in the null space of $\\Delta_{X_M} = (d_{X_M}+\\delta_{X_M})^2$), and the cohomology groups "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.2687","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:22:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a+FcDAUf/aNoRGhW+bia6efwQ/2/8AgLHNxCTwVNk2I/9/6Moi9vzjGPlya4nbZ0KjsKdkNLr3V000mrrH/YDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T11:07:45.546244Z"},"content_sha256":"b893bae552eb533774ef1dd4bbb05e0ba58479a7ee465ceb06b804f143729dfe","schema_version":"1.0","event_id":"sha256:b893bae552eb533774ef1dd4bbb05e0ba58479a7ee465ceb06b804f143729dfe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZLHTANQBMBJYUD2DHGUKF6JAGK/bundle.json","state_url":"https://pith.science/pith/ZLHTANQBMBJYUD2DHGUKF6JAGK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZLHTANQBMBJYUD2DHGUKF6JAGK/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-28T11:07:45Z","links":{"resolver":"https://pith.science/pith/ZLHTANQBMBJYUD2DHGUKF6JAGK","bundle":"https://pith.science/pith/ZLHTANQBMBJYUD2DHGUKF6JAGK/bundle.json","state":"https://pith.science/pith/ZLHTANQBMBJYUD2DHGUKF6JAGK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZLHTANQBMBJYUD2DHGUKF6JAGK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:ZLHTANQBMBJYUD2DHGUKF6JAGK","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f4c323d90815eae92fe484b859f1119678a7566c62860af42c9199489f10b7f8","cross_cats_sorted":["math-ph","math.AT","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-04-15T18:51:51Z","title_canon_sha256":"ef374625b534d69b54fdce8bcb83ecb892b455d313ceed8f88f275b810c6ff4e"},"schema_version":"1.0","source":{"id":"1004.2687","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.2687","created_at":"2026-05-18T04:22:48Z"},{"alias_kind":"arxiv_version","alias_value":"1004.2687v3","created_at":"2026-05-18T04:22:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.2687","created_at":"2026-05-18T04:22:48Z"},{"alias_kind":"pith_short_12","alias_value":"ZLHTANQBMBJY","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"ZLHTANQBMBJYUD2D","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"ZLHTANQB","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:b893bae552eb533774ef1dd4bbb05e0ba58479a7ee465ceb06b804f143729dfe","target":"graph","created_at":"2026-05-18T04:22:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We consider a compact, oriented, smooth Riemannian manifold $M$ (with or without boundary) and we suppose $G$ is a torus acting by isometries on $M$. Given $X$ in the Lie algebra and corresponding vector field $X_M$ on $M$, one defines Witten's inhomogeneous coboundary operator $d_{X_M} = d+\\iota_{X_M}: \\Omega_G^\\pm \\to\\Omega_G^\\mp$ (even/odd invariant forms on $M$) and its adjoint $\\delta_{X_M}$. In the 1980s Witten showed that the resulting cohomology classes have $X_M$-harmonic representatives (forms in the null space of $\\Delta_{X_M} = (d_{X_M}+\\delta_{X_M})^2$), and the cohomology groups ","authors_text":"James Montaldi, Qusay S.A. Al-Zamil","cross_cats":["math-ph","math.AT","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-04-15T18:51:51Z","title":"Witten-Hodge theory on manifolds with boundary and equivariant cohomology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.2687","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:377566afc7468ae9b3b7edb580c425fd3d7ef145fadda1087c20a7947538e478","target":"record","created_at":"2026-05-18T04:22:48Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f4c323d90815eae92fe484b859f1119678a7566c62860af42c9199489f10b7f8","cross_cats_sorted":["math-ph","math.AT","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-04-15T18:51:51Z","title_canon_sha256":"ef374625b534d69b54fdce8bcb83ecb892b455d313ceed8f88f275b810c6ff4e"},"schema_version":"1.0","source":{"id":"1004.2687","kind":"arxiv","version":3}},"canonical_sha256":"cacf30360160538a0f4339a8a2f92032b3267cec8c0dad64a27af60a3cf8b707","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cacf30360160538a0f4339a8a2f92032b3267cec8c0dad64a27af60a3cf8b707","first_computed_at":"2026-05-18T04:22:48.846845Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:22:48.846845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L6YtO4n9Sf3+xMffrNk4CK4uRn8z6PFbXgvhKOy02ZJzl5T6VxpReI53ZaldpWcBRH4GUGcFPWPGlDv/L2epBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:22:48.847374Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.2687","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:377566afc7468ae9b3b7edb580c425fd3d7ef145fadda1087c20a7947538e478","sha256:b893bae552eb533774ef1dd4bbb05e0ba58479a7ee465ceb06b804f143729dfe"],"state_sha256":"17b46149bc01b34327edf8dcea25ee6b2c95e1a3600b9ec63ea0d7a414217646"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wg42Sd4INPtuPYekd14CsdG42zA2/ECuAbqxy7mQ7VgHxNDVi000NOdFtxwZGIUNGZN2aj/BgipB8Kshn3dOCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T11:07:45.548074Z","bundle_sha256":"36e111da24cb0b49bd72026e2af0ad5e95e3307f78a78135126e4f714706ea0a"}}